Different types of quasi-weighted αβ-statistical convergence in probability

Abstract

The sequence of random variables {Xn}n?N is said to be weighted modulus ??-statistically convergent in probability to a random variable X [16] if for any ?,? > 0, limn??1 1/T??(n) |{k ? T??(n): tk?(P(|Xk-X|? ?)) ? ?}| = 0 where ? be a modulus function and {tn}n?N be a sequence of real numbers such that limn?? inf tn > 0 and T??(n) = ? k?[?n,?n] tk ? n ? N. In this paper we study a related concept of convergence in which the value 1/ T??(n) is replaced by 1/Cn, for some sequence of real numbers {Cn}n?N such that Cn > 0 8 n ? N, lim n?1 Cn = ? and lim n?1 sup Cn T??(n)< 1 (like [30]). The results are applied to build the probability distribution for quasi-weighted modulus ??-statistical convergence in probability, quasi-weighted modulus ??-strongly Ces?ro convergence in probability, quasi-weighted modulus S??-convergence in probability and quasiweighted modulus N??-convergence in probability. If {Cn}n?N satisfying the condition lim n?1 inf Cn/T??(n) > 0, then quasi-weighted modulus ??-statistical convergence in probability and weighted modulus ??-statistical convergence in probability are equivalent except the condition lim n?1 inf Cn T??(n) = 0. So our main objective is to interpret the above exceptional condition and produce a relational behavior of above mention four convergences.